The eggbox effect: converging the mesh cutoff

The eggbox effect:
converging the mesh cutoff
Objectives
- study the convergence of the energy and the forces with respect to
the real space grid.
System where the tests will be performed:
bulk MgO (rocksalt structure)
Instead of using the unit cell (FCC + 2 atoms of basis), the exercise will be more
transparent if we use a conventional unit cell with orthogonal lattice vectors
Simulation cell:
•Tetragonal
•4 atoms/cell
Some of the integrals are computed in a
three dimensional grid in real space
Let us project the atomic position in the (x,z) plane
Simulation cell:
Mg
•Tetragonal
O
•4 atoms/cell
y=0
z
y=0
x
y=0.5
y=0.5
Three dimensional grid to compute
Hartree, exchange correlation and neutral atom potentials
Find all the atomic orbitals that do not vanish at a given grid point
(in practice, interpolate the radial part from numerical tables)
EVERYTHING O(N)
Once the density is known, we compute the charge density and the potentials
Fineness of the grid controlled by a single parameter,
the “MeshCutoff”
Ecut : maximum kinetic energy of the plane waves that can be
represented in the grid without aliasing
where
is the grid interval
∆x
In the grid, we represent the density ⇒ grid cutoff not directly comparable
with the plane wave cutoff to represent wave functions
(Strictly speaking, the density requires a value four times larger)
Fineness of the grid controlled by a single parameter,
the “MeshCutoff”
MeshCutOff = 100 Ry
Grid of 18 x 18 x 30 points
along the three lattice vectors
Mg
O
y=0
z
y=0
x
y=0.5
y=0.5
All the quantities should be invariant under translation as
a whole, but the grid breaks translation symmetry.
The grid integrals make the energy dependent on the
position of the atoms relative to the grid
Mg
Relative position can be controlled
by the input variable:
O
y=0
y=0.5
%block AtomicCoordinatesOrigin
The origin is given in the same
units as the atomic coordinates.
z
y=0
x
y=0.5
The grid integrals make the energy dependent on
the position of the atoms relative to the grid
MeshCutOff = 100 Ry
Grid of 18 x 18 x 30 points along the
three lattice vectors
Mg
Distance between consecutive points in
the grid along z (in reduced coordinates):
O
y=0
z
y=0
y=0.5
y=0.5
1/30
Let us compute the change in the energy
and the forces when we displace rigidly
all the atoms in the unit cell from one
point of the grid to the next one (let us
assume, in this case, in 10 steps)
%block AtomicCoordinatesOrigin
x
0.0 0.0 (1/30)/10
%endblock AtomicCoordinatesOrigin
The grid integrals make the energy dependent on
the position of the atoms relative to the grid
%block AtomicCoordinatesOrigin
0.0 0.0 1/30/10
Mg
%endblock AtomicCoordinatesOrigin
O
y=0
z
y=0
x
y=0.5
y=0.5
The grid integrals make the energy dependent on
the position of the atoms relative to the grid
%block AtomicCoordinatesOrigin
0.0 0.0 2/30/10
Mg
%endblock AtomicCoordinatesOrigin
O
y=0
z
y=0
x
y=0.5
y=0.5
The grid integrals make the energy dependent on
the position of the atoms relative to the grid
%block AtomicCoordinatesOrigin
0.0 0.0 3/30/10
Mg
%endblock AtomicCoordinatesOrigin
O
y=0
z
y=0
x
y=0.5
y=0.5
The grid integrals make the energy dependent on
the position of the atoms relative to the grid
%block AtomicCoordinatesOrigin
0.0 0.0 10/30/10
Mg
%endblock AtomicCoordinatesOrigin
O
y=0
z
y=0
x
y=0.5
y=0.5
Eggbox effect
The war against the eggbox…
Solution 1: Increase the MeshCutoff
MeshCutOff = 100 Ry
MeshCutOff = 200 Ry
Grid 18 x 18 x 30
Grid 30 x 30 x 36
Extra cost in:
CPU time
Memory
The war against the eggbox…
Solution 2: the Grid-cell sampling
Achieve SCF for a given MeshCutoff and relative positions of the
atoms with respect the grid points.
Freeze in the Density Matrix.
Translate the whole system rigidly by a set of points in a finer mesh.
Recalculate energy, forces, and stresses in the shifted configuration,
using the Density Matrix frozen before (that is, the shifted calculations
are non self-consistent).
Take the average of the energies, forces, and stresses between all the
sampled points.
No extra cost in memory.
It is done only at the end of the SCF iteration, for fixed DM. Only
moderate cost in CPU time.
The war against the eggbox…
Solution 2: the Grid-cell sampling
%block GridCellSampling
0.5 0.0 0.0
0.0 0.5 0.0
0.0 0.0 0.5
%endblock GridCellSampling
The war against the eggbox…
Solution 3: filtering the radial functions
Computer Physics Communications 180, 1134 (2009)
The war against the eggbox…
Solution 3: filtering the radial functions
The war against the eggbox…
Solution 3: filtering the radial functions
The war against the eggbox…
Solution 3: filtering the radial functions
Filtering the radial functions
produces, for this system, similar
results than the grid cell sampling,
although with a reduction in the
CPU time
A fine comparison between the two
methods, would require the study of the
convergence of a relevant quantity (for
instance, the frequence of a phonon)
with respect the cutoff, evaluating the
computational cost of each method